Science in Britain looks in rude health. Three hundred and fifty years after the birth of the Royal Society, the UK is ranked second in the world behind the US for the quality of its scientific research on several measures, the number of Nobel laureates it has produced or the number of universities (four) in the world's top 10.

And science seems to be in the news now more than ever – just look at the exciting announcement of the creation of an artificial organism, speculation over the discovery of new subatomic particles at the Large Hadron Collider or the debates over energy production and geoengineering to tackle climate change; and technological innovation is advancing so rapidly that it is sometimes hard to keep up.

Yet there is a puzzling contradiction in the public's attitudes to science. On one hand, recent science documentaries on television have been hugely successful, with physicist and presenter Brian Cox doing the chatshow rounds with an ease that suggests an overdue convergence of CP Snow's Two Cultures. Along with popular science books and magazines, websites and blogs talking about everything from black holes to quantum physics, it seems science is at last entering genuine popular culture. On the other hand, there is a worrying and growing trend of mistrust of science, alongside a resurgence in belief in creationism, and enduring widespread and unquestioning trust in alternative therapies and, most worrying of all, a rise in the number of climate-change deniers.

The proportion of people in the UK, questioned in a survey by Ipsos Mori a few months ago, who believe climate change is "definitely" a reality dropped from 44% to 31% in a single year – apparently owing to a combination of heavy snow last winter and the University of East Anglia email debacle. More troubling still is the knowledge that several members of our new coalition government are climate-change deniers (and I prefer the more accurate term "deniers" to describe such people to the for more accommodating "sceptics", which I reserve for the way good science is meant to work, relying as it does on rational scepticism).

Of course the struggle between the forces of rationalism and irrationalism is age old, and part of what defines us as human. When I hear a moon-landing denier or creationist arguing that the universe is a few thousand years old, or a homeopathy advocate, I smile (not too smugly, I hope), take a deep breath and try to explain how evidence-based, scientific, rational inquiry works. If I cannot convince them, so be it. It was nevertheless a relief to hear British Medical Association leaders recently liken homeopathy to "witchcraft".

There will always be those who deny the evidence, no matter how overwhelming, and while a wider public backlash against the harsh realities of climate change is worrying, I do not see this as a new trend of irrationalism. Scientists will just have to make their case with more honesty and clarity. As well as telling the inspiring stories of scientific advances, the wonders of the cosmos or the mysteries of the atom, we also have a responsibility to explain how science works as a process.

The cliches about science are well known: that it is about having an open mind, never about certainties; that the scientist is not a person who gives the right answers but one who asks the right questions; that the scientific method is the best way of understanding the world, and so on. The fact that irrational and antiscientific attitudes remain so widespread suggests that the case for science still needs to be made.

In making that case, we should feel confident enough to discard any notion that mainstream is the enemy of rigour, or that primetime equals dumbing down. If that means the science programme Genius of Britain goes head to head on the schedules with a very different kind of talent quest, Britain's Got Talent, then bring it on. If science is to genuinely take its place within popular culture on our TV screens then it will have to compete for the public's attention along with everything else.

Jim Al-Khalili is a presenter on Genius of Britain, from Sunday on Channel 4